League One basement club Port Vale will host Chelsea following their stunning victory over Sunderland.
The remaining tie will see Leeds visit either West Ham or Brentford, who play their fifth-round fixture on Monday night.
The headline contest is City's latest encounter with Liverpool. Pep Guardiola's side have already claimed two victories over the Reds this season, winning 3-0 in Manchester in November and 2-1 at Anfield in February.
It will be the clubs' first FA Cup meeting since the 2022 semi-final, when Liverpool edged a 3-2 victory on their way to lifting the trophy.
Arne Slot's side reached the last eight with a 3-1 win at Wolves and are eight-time winners of the competition.
City, who defeated Newcastle in the fifth round, have won the FA Cup seven times and last lifted it in 2023.
Arsenal, who are currently leading the Premier League, progressed with a nervy 2-1 win at League One Mansfield.
Mikel Arteta's side now travel to Championship promotion contenders Southampton, who produced a surprise 1-0 win at Fulham in the previous round.
All ties will be played across the first weekend of April.
FA Cup quarter-final draw in full:
West Ham or Brentford vs Leeds